Applying this rule, we have the following table:—
| Original Proposition | Obverse | Partial Contrapositive | Full Contrapositive |
| All S is P. A. | No S is not-P. E. | No not-P is S. E. | All not-P is not-S A. |
| Some S is P. I. | Some S is not not-P. O. | (None.) | (None.) |
| No S is P. E. | All S is not-P. A. | Some not-P is S. I. | Some not-P is not not-S. O. |
| Some S is not P. O. | Some S is not-P. I. | Some not-P is S. I. | Some not-P is not not-S. O. |
It will be observed that in the case of A and O, the contrapositive is equivalent to the original proposition, the quantity 136 being unchanged, whereas in the case of E we pass from a universal to a particular.[140] In order to emphasize this difference, and following the analogy of ordinary conversion, the contraposition of A and O has been called simple contraposition, and that of E contraposition per accidens.[141]
[140] In most text-books, no definition of contraposition is given at all, and it may be pointed out that, in the attempt to generalise from special examples, Jevons in his Elementary Lessons in Logic involves himself in difficulties. For the contrapositive of A he gives All not-P is not-S ; O he says has no contrapositive (but only a converse by negation, Some not-P is S); and for the contrapositive of E he gives No P is S. It is impossible to discover any definition of contraposition that can yield these results. Assuming that in contraposition the quality of the proposition is to remain unchanged as in Jevons’s contrapositive of A, then the contrapositive of both E and O is Some not-P is not not-S.
[141] Compare Ueberweg, Logic, § 90.
That I has no contrapositive follows from the inconvertibility of O. For when Some S is P is obverted it becomes a particular negative, and the conversion of this proposition would be necessary in order to render the contraposition of the original proposition possible.
As regards the utility of the investigation as to the inferences that can be drawn from given propositions by the aid of contraposition, De Morgan[142] points out that the recognition that Every not-P is not-S follows from Every S is P, whatever S and P may stand for, renders unnecessary the special proofs that Euclid gives of certain of his theorems.[143]
[142] Syllabus of Logic, p. 32.
[143] It will be found that, taking Euclid’s first book, proposition 6 is obtainable by contraposition from proposition 18, and 19 from 5 and 18 combined; or that 5 can be obtained by contraposition from 19, and 18 from 6 and 19. Similar relations subsist between propositions 4, 8, 24, and 25; and, again, between axiom 12 and propositions 16, 28, and 29. Other examples might be taken from Euclid’s later books. In some of the cases the logical relations in which the propositions stand to one another are obvious; in other cases some supplementary steps are necessary.
In consequence of his dislike of negative terms Sigwart regards the passage from All S is P to No not-P is S as an artificial perversion. But he recognises the value of the inference from If anything is S it is P to If anything is not P it is not S. This distinction seems to be little more than verbal. It is to 137 be observed that we can avoid the use of negative terms without having recourse to the conditional form of proposition: for example, Whatever is S is P, therefore, Whatever is not P is not S ; Anything that is S is P, therefore, Anything that is not P is not S.